Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{ m}(z) = \prod_{ k=1}^{ m}{ (z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{ N+1}(z) = p_{ N}(z) \left(z - z^*\right)\qquad \mbox{ where}~z^* = \arg\max_{ |z|=1} |p_{ N}(z)|,$$ and prove that, independently of the initial polynomial $p_m$, the roots of $p_{ N}$ equidistribute in angle at rate at most $(\log{ N})^2/N$. This even persists when sometimes adding `adversarial' points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian $(-\Delta)^{ -1/2}$ and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.
Categories: math.CA math.CV